Law of corresponding states for open collaborations

We study the relation between number of contributors and product size in Wikipedia and GitHub. In contrast to traditional production, this is strongly probabilistic, but is characterized by two quantitative nonlinear laws: a power-law bound to product size for increasing number of contributors, and the universal collapse of rescaled distributions. A variant of the random-energy model shows that both laws are due to the heterogeneity of contributors, and displays an intriguing finite-size scaling property with no equivalent in standard systems. The analysis uncovers the right intensive densities, enabling the comparison of projects with different numbers of contributors on equal grounds. We use this property to expose the detrimental effects of conflicting interactions in Wikipedia.

Figure 1

The relation between effort and project size is probabilistic and shows the presence of sublinear bounds. Both features are demonstrated by scatterplots of the number of contributors versus the project size in Wikipedia pages (a) and GitHub software (b). The thick dashed lines show the average bounds from several realizations of the model described in the text (examples are shown in the insets): dotted lines are reference power laws fitted from the data (we used the procedure developed in [32] for smoothing out the roughness and fitting the power-law exponent of the bound).

Figure 2

The number of edits per contributor (a) and the total number of contributors per project (b) follow fat-tailed distributions. Wikipedia edits (circles) are aggregated for all pages, while GitHub edits (lines) are shown separately for the ten projects with most contributors (shifted upwards by a factor 100 to increase readability). Dotted lines show reference power laws with exponents $\alpha +1$ (left panel) and $\beta$ (right panel) fitted from data: ${\alpha }_{\mathrm{w}}=1.1\left(1\right),{\alpha }_{\mathrm{g}}=0.6\left(1\right)$, and ${\beta }_{\mathrm{w}}={\beta }_{\mathrm{g}}=2.4\left(1\right)$.

Figure 3

The marginal size distributions at fixed number of contributors collapse onto universal curves, characterizing Wikipedia (a) and GitHub (b). The power-law rescaling yielding the best collapse is given by $n$ raised to the exponent $0.8$ for Wikipedia and $1.7$ for GitHub. These exponents (obtained by a minimization procedure [37]) are equal to the inverse of the activity exponent $\alpha$, as predicted by the model.

Figure 4

The exponent $\gamma$ of the bound depends on the exponent $\alpha$ of the activity distribution. Open symbols are fits from simulated data (Pareto variables in the main plot, $\alpha$-stable in the inset), for $\beta =0$ (red squares), $\beta =0.8$ (blue triangles), and $\beta =2.4$ (magenta rhombi). The dashed lines are the theoretical estimates for stable variables. The solid red circle and blue pentagon are the empirical values in Wikipedia and GitHub: ${\alpha }_{\mathrm{w}}$ and ${\alpha }_{\mathrm{g}}$ as in Fig. 2, and ${\gamma }_{\mathrm{w}}=0.65\left(5\right),{\gamma }_{\mathrm{g}}=0.4\left(1\right)$. The shaded region displays the allowed values of $\gamma$ for the empirical case $\beta =2.4$ (i.e., those lying between the case of Pareto variables and the asymptotic result, $\gamma =\alpha$ for $\alpha <1$, and $\gamma =1$ for $\alpha >1$).

Figure 5

Controversial Wikipedia pages are biased towards small sizes. Circles in the contributors/size scatterplot (a) are shaded according to the conflictuality $M$ (lighter red corresponds to larger $M$). Panel (b) compares the reduced-size distributions (see Fig. 3) of all pages (dashed line) with those of conflictual pages in four ranges of the controversiality index $M$ (solid red lines). [Wikipedia data ([30], accessed 12 February 2015) and conflictuality values (obtained from [49], accessed 12 February 2015) were matched by page title.]

Figure 6

Subtractive contributions affect the bound mildly. Their main effect is the blurring of the bound (red lines) because of the larger variance, which increases with increasing deletion probability; a secondary effect is a small readjustment of the slope. The points are the Wikipedia data set.